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Beliebt Rechnen >

integral of 3x^2ln(x^2+3)

Lösung

\int 3 x^{2} ln (x^{2} + 3) d x

Lösung

x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3}) + C

Schritte zur Lösung

\int 3 x^{2} ln (x^{2} + 3) d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x^{2} ln (x^{2} + 3) d x

Wende die partielle Integration an

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - \int \frac{2 x ^{4}}{3 ( x ^{2} + 3 )} d x)

\int \frac{2 x ^{4}}{3 ( x ^{2} + 3 )} d x = \frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3})

= 3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - (\frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3})))

Vereinfache

3 (\frac{1}{3} x^{3} ln (x^{2} + 3) - (\frac{2 x ^{3}}{9} - 2 x + 23 arctan (\frac{x}{3}))) : x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3})

= x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3})

Füge eine Konstante zur Lösung hinzu

= x^{3} ln (x^{2} + 3) - \frac{2 x ^{3}}{3} + 6 x - 63 arctan (\frac{x}{3}) + C

Graph

Beliebte Beispiele

integral of x(2x-1)^2

\int x (2 x - 1)^{2} d x

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